Design of a Multicore Sparse Cholesky Factorization Using DAGs

The rapid emergence of multicore machines has led to the need to design new algorithms that are efficient on these architectures. Here, we consider the solution of sparse symmetric positive-definite linear systems by Cholesky factorization. We were motivated by the successful division of the computation in the dense case into tasks on blocks and use of a task manager to exploit all the parallelism that is available between these tasks, whose dependencies may be represented by a directed acyclic graph (DAG). Our sparse algorithm is built on the assembly tree and subdivides the work at each node into tasks on blocks of the Cholesky factor. The dependencies between these tasks may again be represented by a DAG. To limit memory requirements, blocks are updated directly rather than through generated-element matrices. Our algorithm is implemented within a new efficient and portable solver HSL_MA87. It is written in Fortran 95 plus OpenMP and is available as part of the software library HSL. Using problems arising from a range of applications, we present experimental results that support our design choices and demonstrate that HSL_MA87 obtains good serial and parallel times on our 8-core test machines. Comparisons are made with existing modern solvers and show that HSL_MA87 performs well, particularly in the case of very large problems.